What Are Classic Beginner's Proofs for Calculus Enthusiasts?

[Null_]

I'm not using the suggested formatting because my question doesn't fit it. If this belongs elsewhere, I apologize for posting it here.

I am currently in Calc II and we are starting to prove certain things. I found out that I really enjoy this and want to do a few proofs a week just for fun. Is there a list of common things to prove, made with a beginner in mind? There are many "classic" math problems, so I assume there are "classic" beginner's proofs as well. So far I've started with just the quadratic formula and Pythagorean theorem. When I get stuck I look online.

Thanks.

 

[micromass]

I think you may get more answers if you posted this thread in the general mathematics forum. You should probably ask a mentor to move your thread there (just hit the report button and ask it).

Anyway, to answer your question. It's a little hard to recommend some proofs to you without knowing what you've seen so far. So could you possibly list the things you've seen or are seeing?

Some classical proofs that I could recommend are:
- prove that \sqrt{2} is irrational.
- prove that e is irrational (you'll probably want some hints for this).
- proving functions to be continuous by using \epsilon-\delta definitions. (this is very tedious, but it's a very important skill!)
- A very fun one, just using trigonometry: prove
\sin(x)+\sin(2x)+\sin(3x)+...+\sin(nx)\leq \frac{1}{\sin(x/2)}
for x\in ]0,2\pi[

If you give me some more information (what kind of proofs you'd like to do, what you've already seen, etc.), then I could give you some more

 

[Robert1986]

I'm not using the suggested formatting because my question doesn't fit it. If this belongs elsewhere, I apologize for posting it here.

I am currently in Calc II and we are starting to prove certain things. I found out that I really enjoy this and want to do a few proofs a week just for fun. Is there a list of common things to prove, made with a beginner in mind? There are many "classic" math problems, so I assume there are "classic" beginner's proofs as well. So far I've started with just the quadratic formula and Pythagorean theorem. When I get stuck I look online.

Thanks.

First, change your major to Math.

Have you been able to prove either of the two listed above?


I assume that your Calc book has proof of certain theorems, doesn't it? Well, just don't read the proofs, try to do them yourself.


Also, how much do you know about modular arithmetic? You could try proving Fermat's Little Theorem (not to be confused with his last theorem.)

 

[Null_]

I'll try those; they look fun!

So far, I've proved that .9 repeating = 1, the distance formula (I did this when doing the pythagorean..it probably isn't formal), the quotient rule, and the product rule. I did the last two while sitting in Calc I learning about them. We have done about 10 formal proofs this year in honors calc II, but I don't count those because the professor just stands at the board and asks the class for the next step.

Robert, I'm thinking about doing a math minor. :)
I was able to prove the quadratic formula with no help, but I did have to get a few hints with the pythagorean theorem.

Good call on looking in my textbook.

I don't even know what modular arithmetic is, but after heading to wikipedia, it seems that I have never encountered it. I'll look into it over spring break.Upon googling, I did find this list: http://www.cut-the-knot.org/proofs/index.shtml . Why did proofs seem so dumb and scary back in 9th grade geometry?

 

[Mark44]

So far, I've proved that .9 repeating = 1, the distance formula (I did this when doing the pythagorean..it probably isn't formal), the quotient rule, and the product rule. I did the last two while sitting in Calc I learning about them. We have done about 10 formal proofs this year in honors calc II, but I don't count those because the professor just stands at the board and asks the class for the next step.

Some others you can think about - proving the identity cos(a - b) = cosa cosb + sina sinb. If I recall, the proof uses the distance formula and the angles are laid out in the unit circle.

Another trig-related one is proving the Law of Cosines.

Upon googling, I did find this list: http://www.cut-the-knot.org/proofs/index.shtml . Why did proofs seem so dumb and scary back in 9th grade geometry?

Doing proofs requires a higher level of mathematical maturity than merely solving equations and the like. You're working with a higher level of abstraction than you are when you substitute numbers into formulas.

There are at least a couple of books devoted just to proofs that might interest you. One is "The Nuts and Bolts of Proofs" by Antonella Cupillari, Wadsworth Publishing Co. There's another that's at my office. I believe the title is "How to Read and Do Proofs." Both are small paperbacks, so the prices are reasonable, I believe.

 

[Null_]

Thanks for the suggestions! My school's library has both of those, so I'll check them out today. :)

 

[micromass]

Maybe this book is something for you: http://store.doverpublications.com/0486613488.html#productdescription
It has all kinds of classic problems with "elementary" proofs (where elementary means that no more than high school math is needed for the solution)

 

[MathAmateur]

"Journey Through Genius" by Dunham is a really great book of classic proofs. I also like "How to Prove it" by Velleman

 

[disregardthat]

You could try doing some high school competition mathematics. They hone your skills in problem solving and proving conjectures you have to make up yourself. This will come in handy when it comes to proving things yourself when you don't know where to start.

When it comes to theorems in calculus and analysis, what you are asked to prove often have a rather specific set of conditions. Remember that in a proof you will most likely use all of these conditions, so ask yourself what information each of these will give you and how they are relevant to what you are asked to prove.